Dynamic Structural and Contact Modeling for a Silicon Hexapod Microrobot

Microscale walking robots, typically with maximum dimensions on the order of a centimeter or smaller, have been proposed or developed over recent decades based on a variety of electromechanical actuation principles, including electrostatic [1], electrothermal [2–4], magnetic [5], shape-memory alloy [6], and piezoelectric [7] transduction. Thin-film piezoelectric ceramic actuators [8], as a class of smart materials for micro-electromechanical systems, share several advantages with bulk piezoelectric materials. These include large work densities and substantial force generation. As thin-film, piezoelectric actuation can also be achieved with modest actuation voltages (typically 5–20 V) and over comparatively large deflections via beam bending. Drawbacks of thin-film piezoelectric microactuators include complicated fabrication requirements, limitations on material compatibility with some processes or other materials, and relative fragility compared to many semiconductor or polymer materials. Recently, it was shown that some degree of fragility and processing complexity could be compensated by integrating lead–zirconate–titanate (PZT) thin-films with high-aspect ratio polymer microstructures and coatings based on parylene-C after PZT deposition [7]. Parylene-C films and microbeams were shown to help protect the fragile piezoelectric layer and amplify actuator stroke in compliant mechanisms, producing more robust thin-film PZT microrobots than previously demonstrated.

A further benefit of thin-film piezoelectric actuation for microrobotic applications is that these actuators can achieve high bandwidths with low damping ratios, which raises possibilities for dynamic or resonant gaits during robot locomotion. However, it has been previously observed that the interaction of elastic resonant behavior in microstructures and impact dynamics between small micro-actuators or microrobotic legs and underlying terrain can give rise to highly varying, complex, and sometimes even nearly chaotic dynamic behavior [9–12]. The authors have previously described a multiple-modes model to explain the dynamics of microrobots with similar actuation principle and materials selection, but at the centimeter-scale where small-scale forces have limited effect on foot–terrain interaction [9]. A preliminary extension to a micro-scale robot was presented in Ref. [13], but this did not address robot foot dynamics, body contact, and model-based performance estimation. The modeling process in this work is guided by previous models for centimeter-scale robots, but converts the system dynamics to a hybrid dynamic model that can account for the distribution of modal motion in two directions for each mode and additional microscale ground interaction phenomena that become significant at microscales. Preliminary modeling based on this approach had shown reasonable agreement between identified robot properties and global robot motion (body vertical and lateral displacements), but model refinements and new validation based on experimental measurements of individual robot foot behavior in this work provide further insight into the influence of individual foot–terrain contacts on cumulative robot locomotion.

There have been several prior studies of fundamental dynamic concepts related to small-scale walking or running locomotion. This includes studies originating in locomotion of biological organisms, such as insects [14], as well as tests of robots intended to operate on these principles, typically at the size-scale of several centimeters [14,15]. These works typically develop a lumped-parameter model for leg dynamics and apply it with relatively simple ground interaction modeling, alternating between firm contact and motion of the robot feet in air. This reflects basic concepts of legged robot dynamics such as their foot–terrain and foot–body interaction, which have some different features when examining dynamics of even smaller, silicon-micromachined robots. Regarding the former, foot–terrain interaction can be characterized regardless the number of legs in a robot, but it is significantly influenced by the scale of robots. At the microscale, nonlinear air damping and adhesion become significant factors that should be modeled accurately to estimate the robot dynamics. The foot–body interaction, meanwhile, in micromachined structures depends significantly on elastic structural resonances. These may exhibit coupling between vibration modes in the legs and the body itself, or at least coupling of resonances in multiple legs that may vary based on the number and location of legs with respect to robot body. Again at scales of several centimeters, elastic vibration has been used to generate piezoelectric robot locomotion, but the further coupling of these dynamics with foot–ground interaction at small-scales has not been studied in detail [10,15–18].

Several contributions toward the understanding of millimeter-scale microrobots are then included in this work. At a high level, while prior works have explored effects of small-scale forces and impacts of individual microrobot legs or mesoscale representations, those works had yet to validate modeling of multilegged dynamic impact behavior on millimeter-scale, resonance-driven microrobot locomotion with significant small-scale force contributions. To effectively capture robot motion, multiple vibration modes are included in this model at all legs: three modes in the robot structure evaluated here, with further extension of modes possible with this modeling technique. Also, for each vibration mode, the direction of motion is considered, so the motion in lateral and vertical directions are coupled, where prior modeling assumed minimal directional coupling between modes. The dynamics of adhesion and rest between the feet and ground are further distinguished for a better understanding of the adhesion influence on microrobot dynamics. From this model, the robot forward motion and individual foot motion of future autonomous robots may be better predicted given specific design parameters and prefabrication analysis.

Model validation during this study is performed using a millimeter-scale microfabricated hexapod robot prototype. Figure 1 shows examples of these microrobots before and after detachment from a silicon chip. To interpret both active and passive microrobot dynamics, a dynamic model is constructed that includes: nonlinear foot–terrain interactions, coupled structural resonances of the robot's actuators and structures, and rigid body motion of the robot chassis with multiple degrees-of-freedom. The microrobots as initially fabricated in the silicon chip are suspended by silicon tethers used to support the microrobots through their fabrication process, which also provide electrical signals for characterization. Several locations on microrobots are measured under different loading conditions to identify robot structural dynamics. For ground interaction testing, the microrobot is detached from its chip, and motion is excited on other surfaces. Since the detaching process breaks the electrical connection to the microrobot, out-of-chip dynamics are measured under external vibratory actuation from a shaker. The robot motion simulated by the dynamic model is compared with the empirical results to evaluate the model with respect to robot translation due to vibratory response of its legs and with respect to small-scale force effects on this motion.

This paper is organized as follows: Section 2 introduces the robot design and details of the millimeter-scale hexapod prototypes. Section 3 presents the dynamic model. Section 4 discusses the experimental results for static, in-chip resonant and out-of-chip resonant responses, and simplifications for parameter identification within the model. The robot dynamic model is validated with the out-of-chip measurement of vertical motion of its feet and body and of average forward motion on the vibrating surface. Section 5 discusses validation testing of the dynamic model with comparisons between experimental and simulated results for certain feasible testing conditions. Section 6 discusses implications for potential future autonomous microrobot locomotion based on observations from the dynamic model. Section 7 presents the conclusions of this work.

An example of the microrobot design used for dynamic testing is shown in Fig. 1, both before and after removal from the silicon wafer in which it is fabricated. Details on robot actuator design, fabrication, and testing have been presented in Refs. [8] and [19]. In brief, the robot consists of a central silicon chassis or body (30 μm thick silicon), surrounded by six nominally identical legs. Each leg contains two actuation elements based on thin-film piezoelectric actuators. The piezoelectric actuators are formed on a silicon dioxide base layer (0.5 μm) with two electrode layers of platinum (0.1 μm and 0.2 μm) on either side of the PZT thin-film layer (1.0 μm). The hip actuator consists of a PZT unimorph constrained to act in lateral contraction [7,20], coupled to a high-aspect-ratio parylene-C microbeam that leverages contraction into in-plane rotation. The knee actuator consists of three PZT unimorphs acting in pure bending to rotate the robot foot out-of-plane. In-plane and out-of-plane actuators are connected in parallel electrically and thus must actuate simultaneously, while legs are addressed either individually or in-pairs from electrical interconnects on the robot body. Conceptually, robot locomotion would be based on alternating actuation of legs, say in a tripod gait, using variations in natural frequency and response time between lateral and vertical motion at the foot to create elliptical-type foot motions. Conceptual foot motion with ground contact, when the PZT elements in each leg are actuated in parallel, is demonstrated with Fig. 2. In a prototypical cycle, the robot foot will spend time both in contact with ground and moving in air within a single actuation cycle, or robot “step,” though substantial variation in behavior will be seen in the presence of contact dynamics and small-scale forces. Tested static amplitudes at the robot foot are about 50–100 μm at actuation voltages of 10–20 V. Much larger foot displacements are possible near resonance, as measured damping ratio for the leg structures is near 0.05, but maximum range-of-motion before failure has not been evaluated.

As fabricated in a silicon chip, serpentine silicon springs or tethers support the robot body and provide electrical signals to the robot legs. This allows characterization of active robot leg motion, but does also influence robot dynamics, as compliance of the robot body or chassis itself is constrained by the tethers, and a rigid body mode of the entire robot oscillating on the tethers is introduced. Sections 3–5 examine the various resonant vibration behaviors that occur across the robot legs and body, and use this information to interpret forward robot locomotion in passive walking on a vibrating field after silicon tethers have been severed, and the robot is permitted to translate freely across a surface.

A model for microrobot dynamics is generated as a hybrid dynamic system. In this process, microrobot dynamics are first studied with finite element analysis (FEA) at the individual leg level. Using FEA results and in-chip characterization of the microrobots, the full dynamic model is built to account for multiple resonant modes of the leg–foot system, nonlinear foot–ground interactions, and body motion in five degrees-of-freedom. The resulting dynamic model includes two parts to completely describe the robot motion. The first part is a leg–foot model with three different dynamic motion modes (“dynamic modes” used here in a hybrid system sense as opposed to “vibration modes” or “resonant modes” associated with the elastic compliance of the robot). These dynamic modes are an in-air mode, an impact mode, and an adhesion mode, to cover possible robot foot motions. The second part is a body model to describe the robot body/chassis motion in five degrees-of-freedom, requiring appropriate motion state transformations between relevant reference systems. The dynamic model is then used to predict and compare robot in-plane and out-of-plane motion when actuated by external vibration from a shaker.

A dynamic model of the microrobot's legs is first constructed in comsol finite element analysis software to understand the frequency response of both hip and knee actuators. Both actuators are simulated to predict the structural resonance. With nominal robot design parameters and fabrication results as reported in Ref. [19], the first lateral resonant frequency of the hip actuators is simulated to be 498 Hz; the first vertical frequency of knee actuators is 3671 Hz. Figure 3 shows the simulated structural deformation of these first two modes of the robot legs. It is worth recalling that both actuators in an individual leg are connected electrically and thus are actuated simultaneously. While it is not possible to stimulate both actuators at their maximum motion amplitude simultaneously with a single-frequency actuation signal, it is possible to actuate the robot leg–foot system various vertical and lateral motion ratios by choosing the actuation frequency carefully. The ratio of vertical to lateral motion at each resonance, as motions of the two actuators are not perfectly orthogonal at their respective resonances, and thus, vertical to lateral motion ratio at other frequencies is a critical factor in generating robot locomotion from this architecture. The vertical to lateral motion ratio was determined with COMSOL to be 1:15 at the first lateral resonance and 17:1 at the first vertical resonance. These ratios are used to model the dynamics of microrobots when associated leg resonances are excited by impact after the robot is detached from the wafer.

Individual robot foot motion in the time domain is distributed into three dynamic or hybrid modes. The first dynamic mode describes the free, in-air motion of a robot leg. This mode shows up in both in-chip and out-of-chip motion. No foot–terrain contact is included in this mode, so the dominant external force from ground interaction is the air damping force at the microscale, i.e., squeeze-film damping. Mode 2 describes the impact phenomena of the robot foot. If the robot foot is not capable of leaving the ground because of adhesion forces or gravity, modes 3 and 4 are used to describe leg motion. Again we note that structural dynamics in each mode originate in the modal vibratory dynamics of the system; as such, when referring to modal dynamics associated with specific resonances, we will refer to “vibration modes,” and to “dynamic modes” when referring to the three modes of motion comprising the hybrid system model.

in which k_i and b_i are the spring constant and damping coefficient for resonance i, normalized by the effective mass of that vibration mode; $xi,j$ and $vi,j$ are the displacement and velocity for the ith vibration mode at jth time step. Vibration mode matrices $Ai$ are constructed independently for vertical and lateral directions based on resonant frequencies with dominant influence in those directions. It should be noted here that the robot motion states in the state space form represents the robot foot motion with respect to the robot body. The displacement and velocity in the lateral (y-) and vertical (z-) directions (respectively, $zi$⁠, $vzi$⁠, $yi$⁠, and $vyi$⁠) are expressed as proportional to a combined modal displacement and velocity, $xi$ and $vi$⁠, using motion coupling coefficients $Cz,i$ and $Cy,i$⁠. The motion coupling coefficients measure the amount of motion in lateral and vertical direction contributed by ith mode of the leg: $Cz$ is the ratio of motion in the vertical direction to the overall motion of the ith mode, and $Cy$ is the ratio of the motion in the lateral direction to the overall motion of the ith mode.

The foot velocities (⁠$vzfg$ and $vyfg$⁠) and displacements (⁠$zfg$ and $yfg$⁠) relative to ground take the body motion into account as well. Equations (13)–(16) in the prior work [9] explain the conversion between velocity and displacement with respect to ground (referred to as $vzfg$⁠, $vyfg$⁠, $zfg$⁠, and $yfg$⁠) versus motion relative to the body (⁠$vzfb$⁠, $vyfb$⁠, $zfb$⁠, and $yfb$⁠, as above).

in which $vzfg$ and $vzb$ are the vertical velocity with respect to ground for the robot foot and robot body, respectively, b_gf and b_gb are the coefficient of squeeze film damping for the robot foot and body, respectively, calculated from their component geometries, air properties, and nominal distance from ground; z_g is the ground height; and $zfg$ and $zb$ are the vertical height of robot foot and robot body.

in which the velocity of robot foot before (⁠$vzfb,o$⁠) and after (⁠$vzfb,f$⁠) impact are related to shaker ground velocity before (⁠$vzg,o$⁠) and after (⁠$vzg,f$⁠) impact according to a coefficient of restitution, $cr$⁠. In Eq. (7), the robot foot velocity is relative to the fixed ground condition, which is calculated from robot body motion and foot motion with respect to the robot body, from Eq. (5).

In which the total velocities are related to the dominant modes in their respective directions, the first mode for vertical velocity and the second mode for lateral velocity. Displacement is distributed into these two modes through a similar calculation. In other words, instead of a force calculation based on normal or frictional forces, state are directly updated when impact occurs.

in which t is the contact duration; c_ad,t and c_ad are coefficients for adhesion force. If net upward force on the robot foot is smaller than threshold adhesion force, then the foot velocity (⁠$vfg,j$⁠) and displacement (⁠$xfg,j$⁠) in both vertical and lateral directions are enforced to be same as ground motion. Then, the foot motion (⁠$vfb,j$ and $xfb,j$⁠) relative to body is inferred from body motion. To calculate the motion states (⁠$Xi,j$⁠), only the resonant mode with lowest frequency is actuated at mode 3, as identified in Refs. [21] and [22].

in which K_i and B_i are spring constant and damping coefficient of the ith mode, in this case not normalized by effective mass. The robot will remain in mode 3 if either of these two inequalities is fulfilled. Once a time step is determined to remain in mode 3, the robot foot displacement and velocity are forced equal to the ground condition during that time step.

The last mode for a robot foot is a rest or normal contact mode. In this mode, the robot foot rests on the ground with a positive normal force, because the net force on the foot is downward. The dynamics of this mode are identical to those of mode 3. However, it is useful to distinguish this mode during simulation for the further study on the influence from adhesion force, so that behavior specific to the presence of adhesive forces can be easily assessed. In the dynamic model, if a robot foot would move below ground for a previous step while in modes 2, 3, or 4, the foot's dynamics remain in mode 4.

In Eq. (13), m is the mass of the robot body, and $Ix$⁠, $Iy$⁠, and $Iz$ are the moments of inertia in the x-, y-, and z-directions. $Fdb$ and $Fsb$ are the linear air drag and squeeze-film air damping terms of the robot body from Eq. (6); F_yl and F_zl are the force from each robot leg to body; x_j, y_j, and z_j are the distances between the connection point of the body and the jth leg to the position of the robot center of mass in the x-, y-, and z directions, respectively. Because of the finite gap between ground and the robot body, it is also possible that robot body can have contact with ground if the downward motion of body is large. This contact between robot body and ground is modeled with a coefficient of restitution, as for foot impact in the vertical direction.

The foot–terrain interaction is calculated based on the foot velocity with respect to ground as described under mode 2 of the leg model. The body–foot interaction is calculated from the foot velocity and displacement with respect to the robot body. Therefore, the transformation between these two velocities is important during simulation of body dynamics, which was shown in the previous work [9] as mentioned above.

While initial FEA modeling provides a nominal set of parameters for model parameters associated with specific resonance modes (i.e., effective mass and stiffness parameters), completed robots may deviate from the model due to fabrication nonidealities, and a number of parameters associated with damping and external forces must be identified empirically. This section discusses dynamic frequency-domain testing.

Both in-chip and out-of-chip dynamics of microrobots are measured, as excited by internal piezoelectric and external vibration stimuli, respectively. The experimental setup for in-chip characterization is shown in Fig. 4 (left). A LabVIEW frequency sweep program interfaces a power supply to two micromanipulator probes, which apply the resulting voltage input to bond pads at the base of the silicon tethers to the piezoelectric actuators. A laser Doppler vibrometer or LDV (Polytec OFV 3001 S Controller & 303 Sensor head), and a stereoscope are used to acquire out-of-plane velocity and in-plane displacement measurements, respectively. The test setup for out-of-chip dynamics testing is shown in Fig. 4 (right). Data acquisition is a performed as above, while a shaker (BK Vibration Exciter Type 4809) is used to excite motion of the detached microrobot in a Teflon tray. The out-of-chip motion is measured with both the LDV and camera. The LDV measurement is used to understand the vertical motion of robot, and the video recorded with the camera is used to characterize the lateral speed of the robot. It is worth noting that the quality of the LDV measurement is dependent on the size and materials at the surface of the robot at various measurement points. As the polymer structures forming the foot do not reflect the LDV laser effectively, the motion of parylene-C foot cannot be directly measured. Thus, all experimental measurements of robot leg behavior are taken at least a short distance from the foot itself, for instance at the outer edge of the robot knee joint. When estimating foot displacement or relating simulated foot motion to experimental results, the displacement of the foot relative to the robot body is assumed to scale from measurements taken near the knee according to the geometries of mode shapes generated by FEA.

First, functionality of microrobot actuators during in-chip testing was assessed visually. Static displacement of individual robot feet was measured to be on the order of tens of micrometers using a 15 Vpp step input at 1 Hz. This step amplitude is observable with microscope. Therefore, it is convenient to identify the actuator functionality with a low frequency input (1 Hz or lower). Then, microrobot in-chip characterization is focused on characterizing its dynamics under different frequencies and loading conditions.

Measurements of robot motion were taken at various locations on the robots during swept-sine excitation of the robot legs to verify and refine resonance behavior predicted by the FEA model. It was known that the silicon tether that connects the microrobots to the wafer could also have a large influence on robot dynamics by introducing a mass-spring rigid body vibration mode to the system and stiffening the chassis structure; this was evaluated in part by examining the effects of constraining the vertical motion of the robot chassis with a micromanipulator probe.

Figure 5 plots the frequency response of out-of-plane velocity measured by the LDV at the robot chassis, at the robot “hip” (just after the in-plane actuator), and at the robot “knee” (just after the vertical actuator). Although the elastic structure of the robot allows resonances to be transmitted throughout its structure, comparison of locations at which various resonances become most significant allows their sources to be verified. Testing results indicate that the robot chassis rigid body resonance on the tethers occurs around 830 Hz; the hip or primary lateral resonance is around 438 Hz (versus 498 Hz from COMSOL); the knee or primary vertical resonance is around 3.4 kHz (versus 3671 Hz from COMSOL). The FEA and measured results matched well for both actuators in the frequency domain. The body or chassis resonance at 830 Hz was confirmed as a rigid body mode through comparison of velocities at additional locations on the chassis, showing nearly pure translation. These measurements also confirmed that elastic modes of the chassis itself are small below 3 kHz when constrained by the silicon tethers. Further confirmation was provided by adding additional stiffness supporting the robot with a micromanipulator probe, which increased the associated resonant peak from 830 Hz to 1.8 kHz, with negligible effect on the modes attributed to the individual legs.

One significant difference between FEA modeling and experimental testing was observed with regard to vertical motion of the knee or out-of-plane actuator. The end points of the three unimorph PZT beams comprising the knee actuator were measured and found to exhibit unequal velocity amplitudes, which indicates the existence of significant foot off-axis tilting motion. This had previously been predicted to be small during FEA analysis. While this was not intentionally designed for microrobot locomotion, such motion has potential to be beneficial if it can supplement or be used in-place of originally designed in-plane foot rotation originating in the hip actuator.

Since the eventual goal of microrobot motion is to realize autonomous, after characterization of in-chip dynamics one robot was detached to evaluate dynamic response during interaction with ground. As extracting the robot at this stage in development removes access to an active actuation signal, motion for out-of-chip testing was excited by a shaker. Figure 6 (left) shows the absolute motion amplitude of the tray, which is used as a reference to calculate the relative motion of three locations on the robot (chassis, hip, and knee), as shown in Fig. 6 (right) under small amplitude ground excitation. It should be noted that the vibration velocity from the shaker decreases with frequency amplitude of robot motion is often small, such that only velocity measurements under about 3 kHz could be clearly distinguished from background noise. Unfortunately, given a finite availability of amplification settings for the current supply to the shaker, while a frequency sweep versus input voltage can be readily tested, only a small range of frequencies permit multiple ground amplitudes to be excited at the same frequency, and minimal robot motion could be observed at those settings.

In Fig. 6 (right), the relative velocities of three locations on the robots with respect to ground motion are plotted versus frequency. The largest motion amplitude is observed between 1.5 and 2.5 kHz. This mode is predominantly in the vertical direction as observable in the experimental setup. It is taken to be the primary vertical vibration leg of the mode after release from the wafer, being reduced from in-chip testing due to additional elastic chassis bending that is constrained by the silicon tethers when they were present in-chip. Additional vertical motion is measured around 800 Hz and 1.2 kHz, believed to be a resonance of body motion itself since these modes were not measured when a probe was pressed to the robot body in the chip.

The overall conclusion of frequency response testing was that after extraction from the wafer, the most significant resonant behaviors were present near 438 Hz, 1.2 kHz, and 1.8 kHz, attributed to vibration modes associated with in-plane rotation at the hip, foot tilting motion, and out-of-plane leg bending due to compliance through the released body, leg, and knee, respectively. To simplify the dynamic model, an additional 800 Hz resonance was not considered in simulation, because it was only found with in-chip testing, and attributed to robot oscillation on its tethers. The resonant frequencies and amplitudes obtained were used to tune stiffness, mass, and input parameters in the state-space model for robot structural dynamics.

The squeeze film damping coefficients are calculated for both robot body and foot, as shown in Table 1, from models in Refs. [21] and [22]. The coefficients for time-dependent adhesion force are estimated from the previous work [23].

Using the identified system model and parameters generated through the experiments described in Secs. 3 and 4, the effectiveness of the proposed dynamic model at describing global robot motion in the presence of foot–ground interaction could be examined. The parameters used in simulation are listed in Table 2, including the mass of robot, resonant parameters. Two primary sets of data are available for model assessment: measurements of average lateral speed of the robot for various vertical ground excitation frequencies, and detailed time-domain measurements of vertical robot leg displacement, measured at the knee or hip.

At key frequencies, the detached hexapod microrobot was found capable of fast lateral motion within the tray when excited by vertical vibration. Figure 7 shows two frames of sample motion recorded when the shaker was operating at 240 Hz with an amplitude of 25 μm. Sample robot motion video stills are shown with a time separation of 2 s. The speed of the robot is about 4 mm/s, with a small counter-clockwise turn observed, possibly due to one robot leg showing signs of damage during the detachment process. Because of the turn, the speed, instead of velocity, of robot is used for further validation of the dynamic model.

The existence of lateral motion of the robot near 300 Hz suggests that its locomotion during ground excitation is generated by external vibration coupling to the hip actuation mode below its resonant frequency, which is also the mode of lateral foot actuation. Simulated robot motion at different frequencies was then compared with experimental measurement. Measured and simulated speeds are shown in Fig. 8. The error bar of the simulation is generated from the uncertainty of robot parameters, and the error of measurement is caused by the nonuniformity of robot motion speed. When the actuation frequency is higher than 400 Hz, no lateral motion was observed. The speed trends for simulation and experiments are in qualitative agreement, showing a fast decline in speed from 200 Hz, and at around 300 Hz there was an increase in robot speed found both in simulation and experiments. The experimental setup (i.e., LDV and stereoscope supports) exhibits resonance when the shaker is actuated at a frequency lower than 150 Hz, preventing lower-frequency characterization using the current setup.

Also shown in Fig. 8, the simulation correctly predicts that no motion will occur at frequency ranges from 500 Hz up to 1.8 kHz, where experimental tests showed no measurable locomotion. This is despite the presence of various vibration modes in that region, but consistent with those vibration modes acting effectively out-of-plane with only small in-plane components. Ability to excite locomotion is also limited at higher frequencies, because maximum amplitudes achievable by the shaker become smaller. The robot speed is mildly overestimated through the entire frequency range, which may indicate the existence of other microscale forces or more complex adhesion behavior beyond the contents of the current model. Nonetheless, the model does accurately estimate the overall speed trend.

More detailed comparison of experimental and model results, and insight into effects of small-scale forces, was obtained from LDV measurements of vertical motion of individual robot legs. Different motion patterns were observed as characteristic for tests at varying frequencies and amplitudes. Three representative patterns are shown in Fig. 9. The first example (Fig. 9 (top-left and top-right)) is a firm-contact pattern, in which the robot foot has constant contact with ground for small ground oscillations. Figure 9 (bottom-left) shows a partial firm-contact pattern at increased amplitudes: the robot stays in adhesion mode for some time length of each period, even beyond the point at which body motion would have otherwise pulled the leg free of the ground, but the foot does detach for an approximately uniform amount of time in each cycle. Figure 9 (bottom-right) is a jumping or bounding pattern when the shaker is actuated at higher amplitudes. Leg motion is stimulated to a much greater extent by the external vibration, and individual legs stay in the air for more than one actuation period, with substantially higher amplitudes (200 μm) than the underlying ground amplitude (25 μm). The appearance of different patterns depends on the actuation condition and initial condition of the robot in tray. The firm-contact pattern is measured when robot is only moving vertically when the shaker is actuated at 4 V peak-to-peak; the jumping pattern is measured at a temporary pause within robot lateral in-tray motions when the shaker is actuated at 8 V peak-to-peak.

Figure 10 compares the simulated vertical motion of robot knee with various external vibrating amplitudes, to support the assumption of distinct motion modes in the hybrid model. With small external vibration, the robot foot is accurately predicted to stay in contact with ground throughout motion. Larger external vibration could take off robot leg and show the in-air mode, in which the robot knee moves with a different velocity than the ground. By comparing them together, examples of adhesion mode behavior are identified. The abrupt velocity change from tracking ground to motion in-air mode is representative of a downward adhesive force that is broken suddenly when sufficient upward force is exerted from the body to the leg being observed.

Figure 11 shows resulting motion profiles of firm-contact, partial firm-contact, and jumping patterns appearing in the simulation. It should be emphasized that these conditions are only observed in simulation results when accounting for the three major out-of-chip vibration modes, external air damping identified from out-of-chip measurement, squeeze-film damping estimated from robot geometry, and adhesion forces estimated from prior works. Figure 11 (top-left) shows the firm-contact pattern when the shaker is actuated with 2 V peak-to-peak, as in experiments. Partial firm-contact starts to appear when the voltage signal increase to 4 V peak-to-peak (top-right plot in Fig. 11). This similar motion pattern is observed in experiments, though it is first observed only above 4 V/below 6 V, again implying some additional adhesion behavior as inferred from robot speed comparisons. For 8 V peak-to-peak actuation (bottom-right plot in Fig. 11), a jumping pattern with minimal adhesion time is observed in both experiments and simulation.

The relative importance of various small-scale forces is also examined through comparison between full and simplified simulations. Sample predictions of foot vertical motion in the presence of various possible small-scale forces are shown in Fig. 12. Both adhesion force and squeeze film damping force are small-scale, nonlinear factors that substantially influence the simulated results for robot leg motion and are critical to generating comparable foot motion patterns to those seen in experiments. Without the presence of a finite adhesion force, the partial firm-contact breaks the similarity between each actuation cycle, and no fully periodic foot motion is generated. It is important to note that “adhesion force” as treated in this work makes no distinction between various intermolecular forces that might influence its magnitude, but if even a small amount of adhesive attraction is present between the robot foot and its terrain, it is predicted to have a substantial influence on robot locomotion. At small locomotion amplitudes, it can prevent effective movement, and at larger locomotion amplitudes, the adhesion force transfers the additional surface motion into the robot's compliant structure, which can be beneficial for forward motion. In contrast, squeeze-film damping reduces all individual leg motions to a substantial degree, so its collective influence is found to reduce robot speed. Therefore, understanding microscale forces are critical for the estimation of robot dynamics.

Based on a dynamic model for structural and contact dynamics, the motion of a piezoelectrically actuated microrobot has been simulated. Microscale forces, such as adhesion and squeeze film damping, are studied to understand their influence on robot locomotion. The robot speed is simulated to be in the scale of mm/s, with predicted speed in agreement with experimental speed in passive locomotion stimulated by vertical ground vibration. Error bars are simulated accounting for the resolution of fabrication and variation in forward motion over the duration of various simulations and are likewise consistent with observed robot locomotion variability on a sample surface.

As a passive walker, this dynamic model can be compared with a smaller and faster robot modeled in Ref. [24]. The model techniques of Ref. [24] have a decent match for free leg motion between experiments and simulation, but are not able to give any prediction on the contact performance and forward motion. Compared to a single leg model from Ref. [12] which does include contact interaction, the leg motion prediction on its own is less accurate, but this work permits estimation of robot forward motion and synthesis of multiple foot contacts, which was not done previously. As a candidate autonomous walking robot technology, comparison of the current robot's range of motion has been done in Ref. [19], with a basic projection of velocity, though findings from this work would suggest a reduction in robot speed and payload capacity due to squeeze-film damping and adhesion.

Toward the goal of predicting requirements and capabilities for dynamic locomotion of such an autonomous microrobot in dynamic locomotion under piezoelectric actuation, we also applied the model developed above to a hypothetical robot being driven by an on-board power supply. To simulate autonomous locomotion, ground height is held constant in the dynamic model, and the internal piezoelectric forcing (⁠$Fact$ in the hybrid system model) are taken to be a square wave around the resonance of the actuators, to produce a tripod gait.

Figure 13 shows the simulated influence of adding robot payload when driving robot motion through on-board actuation near the first vertical resonance. The current microrobot with structure and actuators is about 0.3 mg. In simulation, when the robot total mass is larger than 1.7 mg, the robot stop moving. In some cases, locomotion is predicted to achieve a higher speed with a payload, as it improves uniformity of foot impact during simulation of a tripod gait. This is seen in Fig. 13 as projected robot speed increases beginning when the robot mass is larger than 1.2 mg, because the robot foot has more contact with ground and fewer instantaneous rebound or bouncing impacts. When the total mass of a robot is larger than 1.7 mg, the actuation force is not large enough to support sustained vertical motion on all steps. Around 1.7 mg, the robot speed estimation has a large variation, because the robot can move quickly with lots of small bouncing impact to gain forward speed, or stick to the ground with almost no motion, heavily dependent on initial conditions and random perturbations to the robot actuation force or geometry.

Figure 14 shows an example of robot speed trends as a function of frequency, in this case near the robot vertical resonance (1.8 kHz), associated with primarily lateral motion. This differs from ground excitation measurements, where vertical ground amplitude easily breaks foot-to-ground contact at various frequencies, but high-frequency excitations are both small amplitude (limited by the shaker) and excite only out-of-plane oscillation. In hypothetical untethered locomotion driven by the PZT, sufficient vertical foot displacement to break ground contact occurs most readily near the out-of-plane leg resonance (1.8 kHz), but electrically coupled in-plane actuation is still at least partially excited by the on-chip voltage.

The dynamics of a sample silicon hexapod microrobot are studied as an example for understanding the legged microrobots based on thin-film piezoelectric actuation. These robots have structural dynamics featuring elastic, linear resonances, light damping, and resulting high sensitivity to ground impact interactions. A dynamic model is modified from the previous work with centimeter-scale piezoelectric robot prototypes, further integrating microscale feature such as adhesion and squeeze damping. The model is validated with passive out-of-chip locomotion, both lateral and vertical, of the detached hexapod microrobot, using model information extracted from design parameters, finite element analysis results, and in-chip characterization.

Key features of small-scale motion near resonance examined in this work include squeeze-film damping and adhesion forces, with substantial effects for accuracy of foot motion and forward locomotion predictions, respectively. The model permits simulation of robot motion both with external vibration and on-board electrical actuation, allowing some exploration of potential robot–ground interactions should it operate under on-board electrical power. However, some mismatch between simulation and measurements that can be taken still exists, possibly due to remaining complexities of foot–terrain interaction that are not been fully studied.

Further discussion on the robot motion with on-board electrical actuation was presented to understand possible scenarios for locomotion of microrobots based on thin-film piezoelectric actuation principles. Microscale forces remain critical, but with actuation at vertical resonance, the robot is predicted to move at greater than 1 mm/s with a payload about five times its existing chassis weight, up to about 1.7 mg, when the current mass of the structure and actuators is about 0.3 mg. Payload estimation and the simulation of different actuation frequency around robot lateral resonance emphasize the importance of resonance characterization and operation near one or another of the robot's structural resonances. Further details of other contributing factors to microscale surface interactions, such as electrostatic forces and foot and/or ground plastic deformation, may improve the accuracy of dynamic model prediction. Further analysis will also examine other actuator inputs having potential for sustained efficient locomotion. Future tasks for practical microrobot development include the validation of payload capability and integration with on-board battery or wirelessly coupled power supplies.

The authors thank the Lurie Nanofabrication Facility staff for assistance in development of microfabication processes, and Ms. Hermione Li and Mr. Mayur Birla for additional help with experimental testing and image preparation.

• National Science Foundation (NSF) (Award Nos. CMMI 1435222 and IIS 1208233).